refactor: improve content downloading in docs

[Frando]

Description

based on #2085

• do not start downloads without any known node
• track missing content hashes in the live sync session
• start downloads after content propagation properly
• reenable the sync_big test, let's see what CI says this time (was flakey before)

Notes & open questions

Change checklist

• [ ] Self-review.
• [ ] Documentation updates if relevant.
• [ ] Tests if relevant.